Health insurance mandates in a model with consumer bankruptcy

Abstract

We study insurance take-up choices by consumers who face medical expense risk and who know they can default on medical bills by filing for bankruptcy. For a given bankruptcy system, we explore the total and distributional welfare effects of health insurance mandates compared with the pre-mandate market equilibrium. We consider different combinations of premium subsidies and out-of-insurance penalties, confining attention to budget-neutral policies. We show that when insurance mandates are enforced only through penalties, the efficient take-up level may be incomplete. However, if mandates are also supported with premium subsidies, full insurance coverage is efficient and can also be Pareto improving. Such policies are consistent with the incentive structure for insurance take-up set in the ACA. Pareto improvement is possible because the legal requirement that medical providers dispense acute care on credit, together with the bankruptcy option, yields effective subsidies for medical care utilized by the uninsured. Those subsidy funds, however, can serve the initially uninsured better when they are given as ex ante subsidies on insurance premiums.

Appendix

Proof for Lemma 1

$\frac{\partial \Delta }{\partial w_{i}}=u^{\prime }\left( w_{i}-m\right) -\left( 1-\pi \right) u^{\prime }\left( w_{i}\right) >0$. That is the gain from insurance is increasing monotonically with income. For $w_{L}=X$ :$\Delta <0$, and for $w>X+M:\Delta <0$. Thus, due to the continuity of (1) $\exists \widetilde{w}\in \left( X,X+M\right)$ s.t $\forall w<\widetilde{w} :\Delta <0$ and $\forall w>\widetilde{w}:\Delta >0$. $\square$

Proof for the convexity of $\widetilde{w}(m)$: Convexity of $\widetilde{w}(m)~$implies that $\frac{d\widetilde{w}}{dm}$ is positive and increasing. Equation (4) $\frac{d\widetilde{w}}{dm} =\frac{u^{\prime }\left( \widetilde{w}-m\right) }{u^{\prime }\left( \widetilde{w}-m\right) -\left( 1-\pi \right) u^{\prime }\left( \widetilde{w}\right) }>1$ can be written as

$$\begin{aligned} \frac{d\widetilde{w}}{dm}=\frac{1}{1-\frac{\left( 1-\pi \right) u^{\prime }\left( \widetilde{w}\right) }{u^{\prime }\left( \widetilde{w}-m\right) }} \end{aligned}$$

(8)

The above expression is increasing with m, that is $\frac{d\widetilde{w} }{dm^{2}}>0$ iff $\frac{d\frac{\left( 1-\pi \right) u^{\prime }\left( \widetilde{w}\right) }{u^{\prime }\left( \widetilde{w}-m\right) }}{dm}>0$. Deriving $\frac{d\widetilde{w}}{dm^{2}}$ explicitly we obtain

$$\begin{aligned} \frac{d\frac{\left( 1-\pi \right) u^{\prime }\left( \widetilde{w}\right) }{u^{\prime }\left( \widetilde{w}-m\right) }}{dm}=\left( 1-\pi \right) \frac{\frac{d\widetilde{w}}{dm}u^{{\prime \prime }}\left( \widetilde{w} \right) u^{\prime }\left( \widetilde{w}-m\right) -u^{\prime }\left( \widetilde{w}\right) u^{{\prime \prime }}\left( \widetilde{w}-m\right) \left( \frac{d\widetilde{w}}{dm}-1\right) }{u^{\prime }\left( \widetilde{w} -m\right) ^{2}} \end{aligned}$$

(9)

Applying $\frac{d\widetilde{w}}{dm}=\frac{1}{1-\frac{\left( 1-\pi \right) u^{\prime }\left( \widetilde{w}\right) }{u^{\prime }\left( \widetilde{w} -m\right) }}$ to (9) elaborates it to

$$\begin{aligned} \frac{\left( 1-\pi \right) }{u^{\prime }\left( \widetilde{w}-m\right) ^{2} }\frac{\left[ u^{{\prime \prime }}\left( \widetilde{w}\right) u^{\prime }\left( \widetilde{w}-m\right) ^{2}-u^{\prime }\left( \widetilde{w}\right) ^{2}\left( 1-\pi \right) u^{{\prime \prime }}\left( \widetilde{w}-m\right) \right] }{u^{\prime }\left( \widetilde{w}-m\right) -\left( 1-\pi \right) u^{\prime }\left( \widetilde{w}\right) } \end{aligned}$$

(10)

The sign of (10) is determined by the expression in the brackets. It is negative if

$$\begin{aligned} \left| u^{{\prime \prime }}\left( \widetilde{w}\right) \right| u^{\prime }\left( \widetilde{w}-m\right) ^{2}>u^{\prime }\left( \widetilde{w}\right) ^{2}\left( 1-\pi \right) \left| u^{{\prime \prime } }\left( \widetilde{w}-m\right) \right| \end{aligned}$$

(11)

For (11) to be negative,—implying $\frac{d\widetilde{w}}{dm^{2}}>0$—it is sufficient to have $u^{{\prime \prime }}\left( \widetilde{w}\right) >u^{{\prime \prime }}\left( \widetilde{w}-m\right)$ that is $u^{{{\prime \prime \prime }}}\left( \cdot \right) >0$. $\square$

Proof for Lemma 2:

By (2a) $\frac{dm}{dMC}>0$ for any given uninsured level $\widetilde{w}$. Hence, by (4), $\frac{d\widetilde{w}}{dMC}>0$ that is insurance take-up level is also decreasing with MC. For $MC\rightarrow 0:\frac{d\widetilde{w}}{dm}\rightarrow 1$ and $\frac{dm}{d\widetilde{w}}<1$ hence an intersection between (2a) and (3), denoted $E_{L}\left( \widetilde{w}_{L}^{E},m_{L} ^{E}\right)$, is guaranteed. However, as $\widetilde{w}$ increases $\frac{dm}{d\widetilde{w}}\longrightarrow \infty$, guaranteeing a second intersection $E_{H}\left( \widetilde{w}_{H}^{E},m_{H}^{E}\right)$, where $\widetilde{w}_{L}^{E}<$ $\widetilde{w}_{H}^{E}$ and $m_{L}^{E}<m_{H}^{E}$. $\square$

Proof for Lemma 3:

Any insurance coverage level set by the policy maker, $\overline{w}$ implies a corresponding insurance premiums determined by (2a) and the $m\left( \overline{w}\right)$ curve in Diagram 1.$\forall \overline{w}\notin \left( w_{L}^{E},w_{H}^{E}\right)$ the $m\left( \overline{w}\right)$ curve is below $\varphi \left( \overline{w},m\right)$, implying that for the break-even premiums the marginal consumer $\overline{w}$ would prefer being uninsured, hence $SMU>0$. The converse is true for $\overline{w}\in \left( w_{L}^{E},w_{H}^{E}\right)$ as the $m\left( \overline{w}\right)$ curve is above $\varphi \left( \overline{w},m\right)$, implying the marginal consumer $\overline{w}$ would prefer to be insured for the corresponding market premiums, hence $SMU<0$. $\square$

Proof for Proposition 3:

$\forall \psi ^{pigov}\left( t,s\right) :\left( 1-\pi \right) \left( w_{i}\right) +\pi X-t_{i}=w_{i}-\left( m-s_{i}\right) \Rightarrow t_{i}+s_{i}=\pi \left( M+X-w_{i}\right)$. By Definition 2, all Pigovian incentives equalize expected consumption with and without insurance. Hence, in face of the Pigovian incentive put by any Pigovian policy all risk-averse consumer strictly prefers buying insurance. Then, as everyone is insured $P=MC$ (by 2a). The subsidies cost under Pigovian policy cannot exceed the cost of subsidizing medical care through bankruptcy: $\forall \psi ^{pigov}\left( t,s\right) :$ ${\displaystyle \int \nolimits _{w_{L}}^{w^{E}}} s_{i}di\le \frac{1}{F\left( w^{e}\right) } {\displaystyle \int \nolimits _{w_{L}}^{w^{E}}} p_{i}(M^{E}+X-w_{i})di$ . Hence any Pigovian Policy can indeed be funded through a tax $\tau \le \left( m^{E}-mc\right)$ on the initially insured, without making them drop their insurance. $\square$